CELLULAR BASED DISPATCH POLICIES FOR REAL-TIME VEHICLE ROUTING. February 22, Randolph Hall Boontariga Kaseemson

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1 CELLULAR BASED DISPATCH POLICIES FOR REAL-TIME VEHICLE ROUTING February 22, 2005 Randolph Hall Boontariga Kaseemson Department of Industrial and Systems Engineering University of Southern California Los Angeles, CA

2 ABSTRACT This paper examines a real-time dispatch strategy for picking up small package shipments. Customers make requests for pickups while vehicles are in the field, retrieving shipments from other customers. All requests for pickup must be made prior to a daily cutoff time, and all requests made prior to the cutoff time must be served by the end of the day. We compare cellular based dispatch strategies within fixed territories. That is, a vehicle is routed from cell to cell, rather than stop to stop. Ideally, the vehicle would wait until all requests have arrived before beginning the pick-up process. However, time constraints force the process to begin prior to the cut-off time, resulting in some repetition in serving areas and stops. To minimize the amount of repetition, an efficiency rating is developed for sequencing cells, based on locally maximizing the ratio of work accomplished to time expended. In simulation tests, this method performed well relative to other heuristics. 2

3 INTRODUCTION In this paper, we develop and evaluate a routing scheme that is designed for carriers that serve demand that arrives at random while vehicles are in the field serving customers. In this scenario it is impossible to pre-determine a route, and it is impossible to completely avoid backtracking, as a unit of demand may arrive subsequent to visiting a particular area. The specific intention of the research is to develop a routing methodology for pickup operations of overnight package carriers, but the approach is sufficiently general that it can be applied to routing any commercial vehicle that serves demand that arrives at random while in the field. Pickups typically occur at the end of the work-day, after most or all deliveries have been completed. A critical difference from deliveries is that pickups are not pre-determined by shipments that are loaded in the vehicle. Instead, we assume that they depend on requests that are made while vehicles are in service. The analysis relies on approximations, which are designed to demonstrate fundamental issues that should be considered in sequencing stops for real-time pickup routing, rather than to obtain absolute precision. The methodology is based on a cellular routing scheme. Each vehicle serves a territory, which is divided into a set of cells. Cells are the smallest geographical units that are utilized in constructing routes. For the United States, a cell would correspond to a collection of Zip + 4 postal codes. Zip + 4 generally represents a block face (either odd side or even side). Cellular routing is appropriate, in part, because stop densities are relatively large. Just as arc-routing methods are used for meter-readers and garbage collectors, where stop density is large, cellular routing methods seem appropriate for 3

4 package pickup and delivery, because density is also large. The cellular approach is currently being used by the author in routing algorithms under test at United Parcel Service. Under our routing method, we assume that the driver has discretion to route his or her truck within a cell, but is instructed on the order in which to visit cells. This approach provides a compromise between the traditional method of allowing drivers to route their own trucks, and the need to provide some direction on efficient route construction. The approach also recognizes that drivers are usually most familiar with local routing conditions, and are better able to construct local routes than a computer. In part this is because the underlying geographic databases are not completely accurate, and in part because the databases do not capture all relevant data (e.g., the preferences and priorities of individual customers). The cellular routing problem considered in this paper is modeled as a queueing system with sequence-dependent service times. Specifically, the time to serve a cell declines when it is in close proximity to the preceding cell, thus reducing the travel time component of the service time. In such situations, it is clearly advantageous to utilize a dispatch policy (i.e., queue discipline) that accounts for the relative locations of cells. Another interesting feature of this type of system is that service time tends to be a decreasing function of queue size, since travel distances decline when the density of stops increases. These queueing phenomena were studied in the work of Bertsimas and van Ryzin (1991, 1993a, 1993b) and Gans and van Ryzin (1999). The authors developed performance bounds on stochastic routing problems in which orders arrive by a stationary 4

5 renewal process. The expected performance of simple heuristics was determined, and performance bounds were developed for optimal dispatch policies under heavy traffic conditions. The routing problems faced by overnight carriers differ from the prior research because of cut-off times and associated non-stationarities in demand. Typically carriers do not accept new calls for service after 3:00 or 3:30 p.m., and routes must be completed by 5:00 or 5:30 p.m. During this period, drivers must make a final sweep through their territory to serve all outstanding requests. In the period prior to the cut-off time (i.e., the last time that the carrier will accept a call requesting service), requests continue to arrive while vehicles are in motion. Though it would be more efficient, in terms of route length, to wait until the cut-off time before beginning pick-ups, driver efficiency would suffer. More drivers and vehicles would be needed to serve the demand, and each would only be used for a few hours out of each day. Instead, by spreading work over more hours of the day, fewer vehicles and drivers are needed, with reduced productivity. These phenomena were modeled analytically in Hall (1996). The paper examined the process by which customer queues evolve in this non-stationary environment, and examined how the rate at which stops can be served depends on the queue size. The current paper extends the prior work by developing and testing an operational model for routing vehicles in a dynamic environment, using similar approximations. More general reviews of dynamic routing can be found in Psaraftis (1995) and Bertsimas and Simchi-Levi (1996). It should be noted that much of the research on stochastic vehicle routing does not allow for the dynamic arrival of requests while vehicles are in motion (see, for instance, Gendreau et al, 1995). 5

6 DESCRIPTION OF ROUTING POLICY In this paper we compare, through simulation, alternative cell based dispatching policies. We assume in all cases that vehicles operate within fixed territories, which can be evaluated individually. The system operates as the pickup system for an overnight carrier. However, we focus on an extreme case in which all requests for pickup are dynamic (i.e., no preplanned stops). Shipments originate at random within the territory. Upon serving all stops in the territory, shipments are transported back to a terminal for processing. Requests for service (i.e., pickup) are accepted from time 0 up to time. To meet connecting service, the goal is to finish serving customers by the end of a pickup period, denoted as P >. The ultimate destination is not relevant to the analysis, as it has no effect on the pickup operation. The performance of a route is measured in two ways: (1) total distance traveled, and (2) expected route completion time. By reducing the expected route completion time, the route tends to be less likely to be finished late. It may also be possible to enlarge the service territory of each driver and reduce the total fleet size. Two modes of routing are followed. Prior to cut-off time,, new calls can be received, making it possible for the queue size of requested pickups to increase, and for cells to be visited multiple times. Subsequent to the cut-off time, no new calls are received. At this point a final tour is created within each territory to cover all outstanding orders. We will refer to the period prior to as the dynamic period and the period subsequent to as the static period. 6

7 We assume that drivers are instructed, in real-time, as to the sequence in which cells should be visited, but are not told the sequence in which stops should be visited within a cell or the streets that should be followed when traveling from one stop to the next. A dispatching policy defines the sequence in which a vehicle visits cells. During the dynamic period, the dispatch is a function of the state of the system when a vehicle finishes serving a cell. For the static period, the complete sequence is constructed at the instant that the cut-off time ( ) is reached. A route is defined by a series of steps, each of which has the following four elements: 1) Upon finishing a cell, the states of all cells are inspected 2) For Dynamic: Based on the states, combined with the position of the vehicle relative to the cells, the next cell is selected, and the vehicle proceeds to it. For Static: Vehicle travels to the next cell in the pre-determined sequence. 3) At the time the vehicle enters the next cell, the queue of stops awaiting service is examined, and the driver constructs a route among the stops that satisfy a selection criterion 4) All stops in the queue at time of arrival are served, and the cycle repeats with Step 1. Once the driver arrives in a cell, the set of stops to visit within the cell is fixed. 7

8 If a new call arrives while the vehicle is inside the cell, it will not be served until the next time the vehicle returns to the cell. This simplifies communication with the driver, and avoids backtracking once service commences within a cell. Following this approach, a complete route can be defined as a sequence of stop sets, with each set defined by a cell combined with a set of customer types. Thus the route sequence R is defined as: R = {(z 1,c 1 ),(z 2,c 2 ), } Where: z k is the index of the k th cell visited on the route c k is the set of customer types that are visited in the k th cell visited In our analysis we consider just two types of customers: regular and sporadic. The former have greater potential to generate multiple requests for service and, therefore, it may be unproductive to serve them prior to the cut-off time. When a vehicle enters a cell, it can elect to visit all stops, visit only regular stops or visit only sporadic stops. Several dynamic routing policies are examined in this research: Fixed Cell Sequence: Vehicles follow a fixed sequence of zones, as in Figure 1, traveling from cell to adjacent cell in a cycle that covers all cells in the region. Longest Queue: The vehicle selects the cell with the largest queue of outstanding stops to visit next. Efficiency Rating: The vehicle selects the cell with the highest efficiency score, as described later. 8

9 The innovation in this paper comes from development of the efficiency rating method, which seeks to maximize the productivity of drivers prior to the cut-off time. By accomplishing as much work as possible prior to the cut-off, the route can be completed earlier in the day. This can either be used to improve service, or to increase the number of cells assigned to each driver. PROBLEM SPECIFICATION We first describe the types of problems studied in our simulation experiments, and then provide the models that are employed. In our experiments, vehicle routes are constructed for a rectangular region, defined by the shape parameter r, representing the ratio of the length of the long side of the rectangle to the length of the short side. The fundamental routing issues are the same for territories of all shapes, and rectangular territories facilitate sensitivity analysis. The rectangular region comprises a grid of C x x C y rectangular cells. We assume that stop requests arrive over time by a Poisson process, with arrival rates that are allowed to vary among cells. We also allow arrival rates to vary by time period. Model parameters are defined as follows: A = area of region k = arrival rate of requests in cell k prior to the cut-off time, (calls/time) 9

10 Regular customers are capable of requesting service two or more times during a day, making it possible for the vehicle to combine multiple requests into a single visit. The characteristics are defined by the following parameters: p k = proportion of requests in cell k that come from regular customers R k = number of regular customers in cell k q k (t) = total queue size of customer requests for pickup in cell k at time s k (t) = number of regular stops that are queued at time t (each stop must have at least one request queued). If a new service request comes from a sporadic customer, it is guaranteed to be a stop that is not already queued, because sporadic customers generate at most one request per day. If a request comes from a regular customer, there is a non-zero probability that its stop is already queued. To simplify calculations, we assume that all regular customers are equally likely to generate a request, independent of their current queue size. Based on the above parameters and assumptions, the probability that a new service request comes from a new stop (i.e., one not currently in the queue) is as follows P(new stop) = P(request is sporadic) + P(request is regular)p(regular request is not from customer that is already queued) = [1- p k ] + p k {[R k -s k (t)]/r k } (1) 10

11 The simulation, presented later, models the movement of a vehicle among cells, while accounting for the number of stops and requests queued for service, which are generated by the process described in this section. 11

12 TRAVEL TIME MODELING A route can be viewed as a sequence of steps, each of which consists of travel between a pair of cells plus some amount of time serving the stops within a cell. In our analysis, we are concerned with modeling the time to complete each step, and not the exact stop sequence within each cell or the time to serve individual stops. Travel time is approximated as follows: T jk (t) time needed to serve cell k following cell j, departing cell j at time t = jk + f k (q k (t+ jk ), s k (t+ jk )) + (2) where jk = travel time from centroid of cell j to centroid cell k = random error term s k (t+ jk ) = number of stops queued at time t+ jk f k (q k (t+ jk ), s k (t+ jk )) = average time to serve q k (t+ jk ) requests created by s k (t+ jk ) stops in cell k jk is an approximation to represent the line-haul distance between cells. Although it is expressed as a distance between centroids, more accurate approximations are possible, with a reduction for cells that are in very close proximity. The function f k depends on the requests and stops in cell k s queue, along with characteristics of cell k. The function f() is modeled as the following square-root function, consistent with the literature on lengths of vehicle routes (e.g., Daganzo, 1984). 12

13 f k (q k (t+ jk ), s k (t+ jk )) = k A k s k (t+ jk ) + k q k (t+ jk ) + k s k (t+ jk ) (3) where k = average time per unit distance of A k s k (t+ jk ) (captures both travel metric and speed of travel) k = average time to serve an individual request in cell k k = average time to serve a stop in cell k (not including travel time) A k = area of cell k. The expression is approximate because the square-root relationship is an asymptotic result, which we are applying to routes with a small number of stops. The simpler square-root expression is utilized to keep the model easily tractable. The basic concept of routing based on efficiency ratings does not depend on the specific travel time model. The formulation accounts for the effects of multiple requests for service coming from a single customer, possibly representing different departments at a single address. In this case k represents the time to enter and exit the building, and k represents the additional time to process the requests (e.g., walk within building and process paperwork), which likely occur in different places within the building. In our simulations, is modeled as a normal random variable, with mean 0 and variance proportional to jk + f k (q k (t+ jk ), s k (t+ jk )) V( ) = [ jk + f k (q k (t+ jk ), s k (t+ jk ))] (4) 13

14 The variance model is representative of situations where the times required to serve stops are independent, and proportionate to the mean. An implication is that the coefficient of variation for the travel time is inversely proportionate to the square-root of the mean, leading to increased precision as the mean increases. 14

15 ROUTE INTENSITY Route intensity is introduced in this section as a parameter for characterizing whether or not the route is operating close to a theoretical capacity. The route intensity represents the ratio of the expected time to complete a route, under an optimal routing policy, to the total time available. The total time available exceeds the time to complete the route if the vehicle is idle for a period of time. We develop lower and upper bounds for an approximation on route intensity. These approximate bounds will be used in experiments to measure how close the system is operating to capacity. Let k represent the expected number of requests in zone k at the cut-off time : k = k T min = time to complete a single tour among cell centroids without visiting Stops (i.e., the time for visiting the entire cell sequence without diverting to any stops within the cells. To develop the intensity parameter, suppose that instead of arriving dynamically, all requests arrive simultaneously. Then the stops can be served as a static route, in the minimum possible time. The expected work (measured in time), for such a static route (E(W )), is approximated by the sum of four elements, representing the time to tour cells without stopping, the travel time within cells visiting stops, the time serving individual requests, and the time serving individual stops: K E(W ) min k A k E(S k )+ k k + k E(S k ) (5) k=1 15

16 where E(S k ) is the expected number of distinct stops in cell k that need to be served in a day, with S k being the random variable representing the number of stops in cell k. It should be noted that the approximation overestimates E(W ) somewhat, as E(S k ) is used in place of E( S k ). The error introduced is slight: with a mean of five stops per route, E( S k ) is over-estimated by less than 3% and with 20 stops per route the error drops to less than 1% (based on Poisson distribution). Accounting for regular and sporadic stops, E(S k ) is approximated as follows: E(S k ) (1-p k ) k + R k [1 {1- p k /R k } k ] (6) The second term represents the expected number of regular customers, and accounts for the possibility that multiple requests can come from a regular customer. Because the number of requests, like the number of stops, is a random variable, Equation 6 is approximate. However, the error introduced is also small (no more than 3% when is k is 10 or greater and requests are Poisson distributed). The route intensity,, is bounded by two ratios. The worst case would be if all requests arrived at the cut-off time,, making the time available to serve stops P-. The best case would be if all requests arrive at the start of the pick-up period, making the time available P. The true intensity falls somewhere between the extremes, because some stops can be served prior to the cut-off time, but not as efficiently as they could be served if they all arrived at the cut-off time. Intensity, is thus bounded as follows: E(W )/P < <E(W )/(P- ) (7) 16

17 On-time performance degrades as E(W ) becomes large, because it leads to a larger route intensity and less idle time. 17

18 EFFICIENCY RATING As mentioned earlier, one of the dispatching policies is based on an efficiency rating, which represents the ratio of the amount of work accomplished when visiting a cell to the time entailed in visiting a cell. The work accomplished represents the marginal reduction in E(W ) relative to the alternative of postponing the cell until after time. When a driver finishes serving a cell, all cells will be examined to determine which would be the most efficient to serve next, on the basis of the ratio of marginal reduction in work to time spent serving the cell. Thus, the underlying goal is to maximize efficiency prior to time by minimizing wasted time visiting cells that are better served at a later time. The marginal reduction for serving cell k after cell j at time t, jk (t), is calculated as follows: jk (t) = [drive time reduction] + [request time reduction] + [stop time reduction] k A k { E[S k ]- E[S jk ]} + k E[q k (t+ jk )] + k {E[S k ] - E[S jk ]} (8) where: E[S k ] is the expected number of stops at time in cell k, if cell k is not served between the current time and time. E[S jk ] is the expected number of stops at time in cell k, if the currently queued stops are served after cell j, and 18

19 The marginal reduction in work depends, in part, on the number of regular and sporadic customers in the queue, and in the cell. Because regular customers can generate multiple requests, it may be advantageous to avoid cells that have large concentrations of businesses prior to the cut-off. E[S jk ] equals E[S k ], minus the expected number of stops that would be served if cell k is served next, plus the number of regular stops that would need to be served a second time if cell k is served next (i.e., the regular stop would generate an additional request before time ). E[S jk ] = E[S k ] - E[S k (t+ jk )] + E[S k (t+ jk )][1-e -p [ - (t+ )]/R k k k jk k ], (9) Where: S k (t+ jk ) is the random variable representing the number of total stops in queue at time t+ jk S k (t+ jk ) is the random variable representing the number of regular stops in queue at time t+ jk The efficiency rating is the ratio of jk (t) to the time to serve the cell: e jk (t) = jk (t)/t jk (t) (10) Because route length is a concave function of the number of stops, and expected number of stops is a concave function of the number of requests, e jk (t) is less than one. Values close to one indicate that the vehicle operates close to maximum productivity. Under the policy, the cell with the highest efficiency rating is selected next. In some cases, the efficiency rating for immediately revisiting a cell can be the undefined ratio of 0/0. A 0/0 ratio is interpreted to be infinite, making the cell lowest on the priority list. Note that this dispatch rule accounts for the relative efficiency in serving the cell, as 19

20 well as its proximity to the current cell. This is because the inter-zonal travel time is included in the denominator but not the numerator of the ratio. It favors cells that are nearby, contain mostly sporadic customers, have multiple stops queued, and are unlikely to generate more requests prior to the cut-off time. Some cells are naturally more efficient to serve prior to the cut-off time, due to the number of regular customers, proportion of demand that belongs to regular customers and total demand. To illustrate these relationships, component efficiency measures were computed for driving time and stop time. Because request time is modeled as a linear function, the component efficiency (work accomplished/time expended on processing shipments) is one in all cases. Stop efficiency, on the other hand, is ordinarily less than one because regular stops may need to be revisited (it would also equal one if only sporadic customers are served). And driving efficiency is less than one both because of revisiting stops and because areas may need to be revisited. These are calculated as follows: es k = stop efficiency = {E[S k ]- E[S k ]}/ s k er k = route efficiency = { E[S k ]- E[S k ]}/ s k where s k is the number of stops that are presently ready to be served. The behavior of the component efficiencies is illustrated in Figures 2 to 5. Figure 2 shows stop efficiency for the following case: R k = 5 p = 1 (all regular customers) 20

21 k = 5 (low demand), 10 (medium demand) and 25 (high demand) The proportion arrived in the figure equals q k (t)/ k, which represents the proportion of the day s requests that have arrived at the time the cell is visited. The graph illustrates these relationships: Efficiency is greatest when R k is large relative to k. With more regular stops and constant demand, each regular stop is less likely to generate multiple requests, reducing the likelihood of revisits. As the proportion arrived approaches one, efficiency increases at an increasing rate, especially when k is large relative to R k It should be noted that efficiency can be quite low until such time that most of the shipments have arrived. Though not reflected in the graph, efficiency also grows as p declines. This is because stop efficiency for sporadic customers equals one, as they do not generate multiple requests. Figure 3 shows route efficiency for the same case as Figure 2. Comparison illustrates that route efficiency is less than or equal to stop efficiency, due to the compounding effect of increasing the separation between stops. In the example, efficiency does not reach 50% until 80-90% of shipments have arrived, at which time efficiency increases rapidly. This illustrates that for regular customers, little of the driving work is actually accomplished until very close to the cut-off time. In Figure 4, route efficiency is evaluated for different splits between regular and sporadic customers. When the proportion arrived is less than 50%, the driving efficiency for cells with all 21

22 sporadic customers can be as much as five times larger than the efficiency for cells with all regular customers. Another strategy would be to only serve the sporadic customers in a cell, deferring the regular customers until the cut-off time is reached. This is illustrated in Figure 5, utilizing the same parameters as Figure 4. When the proportion arrived is small, route efficiency is comparable in the two figures, but for larger proportions, the efficiency is considerably larger when only sporadic customers are served. This means that earlier in the day, well before the cut-off time, it could be advantageous to limit service to sporadic customers within a cell, taking advantage of the higher stop efficiency with little loss in driving efficiency. Later in the day, as the cut-off time approaches, it can become advantageous to serve both regular and sporadic customers. Taking all parameters into consideration, the following situations make it more attractive to serve a cell prior to the cut-off time. Long processing times for individual requests, or when stops generate many requests requiring individual processing (thus making stop and driving time efficiency relatively less important). Driving speeds are large and/or the size of the cell is small, making driving efficiency less important. Most shipments come from sporadic customers that are unlikely to generate multiple stop requests. The number of regular customers is large, making it less likely that they generate multiple requests on the same day. 22

23 Requests for service occur early in the day, on average, resulting in a larger proportion of arrived shipments. An additional consideration is that cells that are close to each other should be served together, to avoid unnecessary inter-cell driving distance, as reflected in the parameter jk. 23

24 EXPERIMENTATION Several critical issues are explored through simulation, including the effects of number cells, arrival rate of calls, the proportion of customers that fall in regular and sporadic classifications, and service time parameters. The simulation covered an entire pick-up cycle, beginning with a dynamic period (a 180 minute period prior to cut-off time), and continuing after the cut-off until all pickups were completed. Performance was measured by calculated the time when the final pickup was completed (always after the cut-off time). A series of simulation experiments was completed to compare the three routing heuristics under a variety of scenarios. Method 1 represents a fixed cycle sequence, Method 2 represents a maximum queue size heuristic, and Method 3 is the efficiency rating method. Most of the cases were designed to ensure that the entire route could be completed in about five hours or less. The following parameters were used in experiments: = 3.3 or 6.7 minutes per unit distance = 2 minutes or 0 minutes per request 1 minute or 0 minutes per request R = 3, 10 or 20 regular stops per zone =.5 in all cases = 180 minutes in all cases p =.2,.5 or.8 k = 48 or 72 requests in entire territory (3 to 12 per cell) 24

25 All scenarios were based on a rectangular territory, with total area of size 1, partitioned into either 4 or 16 square cells, with shapes varying as follows: C x x C y = 1 x 4 (M=4, r=4), 2 x 2 (M=4, r=1) C x x C y = 2 x 8 (M=16, r=4), 4 x 4 (M=16, r=1) These data roughly corresponds to actual package pickup routes, with a territory of one square-mile, and a speed on the order of 8 to 16 mph (representing local streets, and including stops at intersections). For each case, 150 runs were completed, producing standard errors for performance measures on the order of 1% of the estimated value. The completion time is shown for each of the three routing methods for 24 cases in Table 1 and Figure 6, representing a total of 10,800 simulation runs. Figure 6 illustrates the relationship between the completion time and E(W ), representing the expected work. It should be noted that expected completion time is always greater than max{ W', and always less than + E(W ), and is approximated by the average of these bounds. The lower bound approximates the average completion time if all work arrived at the start of the pickup period. The upper bound approximate the average completion time if all work arrived at the end of the pickup period (i.e., exactly at the cutoff time). The results (Table 1) are encouraging, in that the efficiency rating method completed routes approximately 9% earlier than the second best method on average, and out performed other methods in nearly all cases. In general, we found that the efficiency method was most advantageous when the total workload fell in a middle range. When demand is very light, all methods perform nearly the same because the queue is always 25

26 nearly zero at the cut-off time. With heavy demand, a fixed cell sequence could perform well because intra-cell distances became less significant. It can also be seen that decreasing the number of regular stops, or increasing the probability that a request comes from a regular stop, decreases the completion time. This is because the expected number of stops is smaller. Concentration of stops within a single cell also lead to an earlier completion time, because clustering stops reduces average distance traveled. 26

27 Table 1. Simulation Results E(finish) by Method k M R p E(W ) 12 1 x x x x x x x x x x x 4* x 2* x x x x x x x x x x x x Average * In this scenario, demand was not evenly distributed. The demand in one cell equalled the demand in three other cells combined. 27

28 CONCLUSIONS AND FUTURE RESEARCH This paper developed the concept of dynamic routing based on relative efficiency. The basic concept is that the driver is routed from cell-to-cell in a manner whereby the next cell visited is always the one that can be served most efficiently. In this manner, the driver can locally maximize his or her productivity prior to the shipment cutoff time, and thus minimize the time that the last pickup is completed. Thus, the method is intended to improve service, and increase the capacity of the driver to handle additional pickups. Efficiency is defined as the ratio of work accomplished to time expended. Work accomplished is the reduction in service time, relative to waiting until all requests arrive. The work accomplished is the reduction in the time serving a cell, relative to waiting until the cut-off time to serve the cell. The tests in this paper are based on a square-root route length approximation for travel distance. However, the basic concept would apply to any model for service time as a function of dynamic cellular characteristics. To implement the method, some modifications are needed to account for stops that must be visited at particular times, based on rigid schedules. These will invariably occur after the cut-off time, during the static portion of the schedule. However, the presence of rigidly scheduled stops could also influence the priorities assigned to cells in the dynamic phase. Additional testing is needed on actual demand distributions and cellular designs. 28

29 REFERENCES Bertsimas, D.J. and Simchi Levi, D. (1996). A new general of vehicle routing research: robust algorithms, addressing uncertainty. Operations Research, 44, Bertsimas, D.J. and van Ryzin, G.J. (1991). A stochastic and dynamic vehicle routing problem in the Euclidean plane, Operations Research, 39, Bertsimas, D.J. and van Ryzin, G.J. (1993). Stochastic and dynamic vehicle routing in the Euclidean plan: the multiple-server, capacitated vehicle case, Operations Research, 41, Bertsimas, D.J. and van Ryzin, G.J. (1993). Stochastic and dynamic vehicle routing with general interarrival and service time distributions, Advances in Applied Probability, 25, Daganzo, C.F. (1984). The distance traveled to visit N points with a maximum of C stops per vehicle: an analytical model and an application. Transportation Science, 18, Gans, N. and G. van Ryzin (1999). Dynamic vehicle dispatching: optimal heavy traffic performance and practical insights, Operations Research, 47, Gendreau, M., Laporte, G. and Seguin, R. (1996). Stochastic vehicle routing, European Journal of Operational Research, 88, Hall, R.W. (1996). Pickup and delivery systems for overnight carriers. Transportation Research, 30, Psaraftis, H.N. (1995). Dynamic vehicle routing: status and prospects. Annals of Operations Research, 61,

30 List of Figures 1. Demonstration of cellular route. 2. Stop efficiency as a function of the proportion of requests that have arrived (low, medium and high demand) 3. Route efficiency as a function of the proportion of requests that have arrived (low, medium and high demand) 4. Routes efficiency for different proportions of regular stops 5. Route efficiency for different proportions of regular stops, serving sporadic only 6. Comparison of Simulation Results to Approximate Bounds 30

31 Figure 1. Demonstration of Cellular Route Stops Remaining in Queue Stops Served 31

32 Figure 2. Stop efficiency as a function of the proportion of requests that have arrived (low, medium and high demand). 32

33 Figure 3. Route efficiency as a function of the proportion of requests that have arrived (low, medium and high demand). 33

34 Figure 4. Routes efficiency for different proportions of regular stops. 34

35 Figure 5. Route efficiency for different proportions of regular stops, serving sporadic only. 35

36 Simulation vs. E(W') E(finish) Method 1 Method 2 Method 3 UB LB Midpoint E(W') Figure 6. Comparison of Simulation Results to Approximate Bounds 36

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